Solving over-determined systems by the subresultant method (with an appendix by Marc Chardin)

Abstract

A general subresultant method is introduced to compute elements of a given ideal with few terms and bounded coefficients. This subresultant method is applied to solve over-determined polynomial systems by either finding a triangular representation of the solution set or by reducing the problem to eigenvalue computation. One of the ingredients of the subresultant method is the computation of a matrix that satisfies certain requirements, called the subresultant properties. Our general framework allows us to use matrices of significantly smaller size than previous methods. We prove that certain previously known matrix constructions, in particular, Macaulay’s, Chardin’s and Jouanolou’s resultant and subresultant matrices possess the subresultant properties. However, these results rely on some assumptions about the regularity of the over-determined system to be solved. The appendix, written by Marc Chardin, contains relevant results on the regularity of $n$ homogeneous forms in $n$ variables.